# What is known about “almost orthogonal vectors”?

Motivation:

Suppose we have a kernel $$k(a,b)$$ defined over the natural numbers.

Then by the Moore–Aronszajn theorem, we can embedd the natural number $$a$$ in some Hilbert space $$\mathbb{H}$$, which we write $$\psi(a)$$.

In the examples I am looking at we have the following:

There are some vectors $$(v_a)_{a \in A}$$, $$v_a \in \mathbb{H}$$ such that:

1. For all $$i,j \in A: \left < v_i, v_j \right > >0$$

2. For all $$\epsilon > 0 \exists i,j \in A: \left < v_i,v_j \right > < \epsilon$$

3. $$\forall a,b \in A: \left < v_a , v_a \right > = \left $$

where $$A$$ is some index set.

In the examples we have $$v_a = \psi(a)$$. It is not difficult to show, that if the vectors $$(v_a)_{a \in A}$$ satisfy 1) and 2) then the index set can not be finite.

I will list the examples I have in mind, some of which are not (yet?) proven to be kernels, but conjectured to be:

1. $$k(a,b) = \dfrac{\gcd(a,b)}{a+b}$$

2. $$k(a,b) = \dfrac{1}{\mathrm{rad}\left(\frac{ab(a+b)}{\gcd(a,b)^3}\right)}$$

3. $$k(a,b) = \dfrac{1}{\sigma\left(\frac{ab}{\gcd(a,b)^2}\right)}$$

4. $$k(a,b) = \dfrac{1}{L\left(\frac{ab}{\gcd(a,b)^2}\right)}$$

where $$\text{rad}=$$ product of prime divisors, $$\sigma=$$ sum of divisors, $$L(n)=H_n + \exp(H_n)\log(H_n)$$, $$H_n=$$ n-th harmonic number.

For 2) and 3) one can use that there are infinitely many primes to get the inquality $$\left < v_a ,v_b \right > < \epsilon$$.

For 4) one can use that the function $$L$$ is strictly monotonically increasing in $$n$$ to get $$\left < v_a ,v_b \right > < \epsilon$$.

Furthermore, if we insist, that $$A = \mathbb{N}$$ then the vectors satisfy a further, fourth property

$$4.\; \forall k,a,b \in \mathbb{N}: \left < v_{ka},v_{kb} \right > = \left < v_{a},v_{b} \right >$$

My question is, if there is anything known about such set of vectors, in this context or related context.

Thanks for your help.

Edit:

Notice also the similarity between the abc-conjecture and the Riemann hypothesis as formulated by Lagarias:

If in the examples 1)-4) all are kernels, then the abc-conjecture and the Lagarias inequality might be written as "kernel-inequlaties":

$$k(a,b) \le K(a,b)$$ or $$k(a,b) < K(a,b)$$

which I find very interesting.

Related question: The abc-conjecture as an inequality for inner-products?